Death penalties are imposed by some governments and not others. Where they are imposed, it is normally for serious offences like murder. Common arguments against the death penalty are

• Killing people costs more than keeping them in prison. This is only because of the administrative burden which we choose to bear. It could be as cheap as one bullet if people had the determination to do so.
• It is wrong to take a human life. Normally this argument has some basis in religion, and those who use it are trying to enforce their religious belief on those around them.
• Killing someone makes you just as bad as they are - you have become a murderer. This argument misses the point that killing an innocent stranger is not regarded the same as killing that stranger's murderer.
• It is not possible to get it 100% right - some innocent people will always die, and some will always be found to be innocent after they have been executed. This is true, but in fact misses the point that the aim is not to ensure that no innocent people are killed, but to minimize the number of innocent people who are killed.

All punishments (including prisons) have a disincentive element. If, by executing one person, ten other would-be-murders are deterred from murder, then there is a nine person gain. Unfortunately

• It is not possible to measure the disincentive effect. Some people feel the effect is very strong, others feel it doesn't exist at all. Any studies are done by biased people who simply choose to use data, or methodologies which supports their belief.
• The knowledge that an innocent person has been occasionally been executed will undermine the belief in the judicial system, which will lead to people justifying their crimes to themselves ('hitting out against an unjust system').
• Seeing the judiciary make mistakes will make some believe they therefore will not be caught.
• Emotionally, people will never accept such an argument with makes the life of one person they can see (the executed) less important that ten 'hypothetical' survivors.

No-one will ever say 'I was one of the people who wasn't mugged and killed because of the disincentive', because they will never know. In fact they may be more scared of being one of the executed innocents - particularly as they will inevitably draw far more mass media attention.